How can I get rid of the link (thick horizontal line) between two keyframes?
The yellow lines do not represent "links" but selected keyframes with the same value.

Form the Blender Manual


  One of the key feature of this window is that it allows you to visualize immediately which channel (i.e. Ipo curve) is really affected. When the value of a given channel does not change at all between two neighboring keyframes, a gray (unselected) or yellow (selected) line is drawn between them."


If what's bothering you is the way the interpolation works between keyframes , you should edit those not the dope sheet but on the Graph editor window.

In this example you can see the same key frames for the location of the X channel represented on the dope sheet and the graph editor, Notice how the Yellow lines between keyframes coincide with the flat regions of the graph, and how the line disappears where there is some transformation on the value in the graph.



EDIT
A note on the Hierarchical system and the Dope Sheet

Everytime you add keyframes using the I key you are presented with an option to choose what channels you want to use for the keyframe, for example Loc,Rot_Scale or Loc_Scale, Rotation only, all channels, only selected channels. The channels you choose are what is called a Keying Set.

The idea behind keying sets is to keep things organized, for example if you don't want to have a location keyframe when you just need to rotate. 

In the DopeSheet you have access both to the keying sets and individual channels and they are organized in a hierarchical structure. You can move, delete or add keyframes with ease.

Here's an example:
At the top of the Hierarchy you'll find the dopesheet summary that encompasses every keyframe on the scene.
On the second level there are the individual objects with their own keyframes.

Click on the trianlges on the left to reveal the next levels down.


The lines rows in green are the Keying Sets



And further down are the individual channels.



Depending on where in the hierarchy you move (or add, or delete) the keyframes, you might affect everything, just the keying set or individual channels.

As for the Yelllow line... Sorry Still no magic nothing there... It doesn't hold anything inside it: It doesn't have any other purpose than to show information in unchanging values between keyframes.